Convective Nozaki-Bekki holes in a long cavityOCT laserSVETLANA SLEPNEVA,1,2,3,* BEN O’SHAUGHNESSY,2,3

ANDREI G. VLADIMIROV,4,5 SERGIO RICA,6,7 EVGENY A.VIKTOROV,8 AND GUILLAUME HUYET1

1Université Côte d’Azur, CNRS, INPHINY, France2Centre for Advanced Photonics and Process Analysis & Department of Physical Sciences, Cork Institute ofTechnology, Cork, Ireland3Tyndall National Institute, University College Cork, Lee Maltings, Dyke Parade, Cork, Ireland4Weierstrass Institute, Mohrenstr 39, Berlin, Germany5Lobachevsky State University of Nizhny Novgorod, 23 Gagarina av., 603950 Nizhny Novgorod, Russia6Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile7UAI Physics Center, Universidad Adolfo Ibáñez, Santiago, Chile8ITMO University, Saint Petersburg, Russia*svetlana.slepneva@gmail.com

Abstract: We show, both experimentally and theoretically, that the loss of coherence of a longcavity optical coherence tomography (OCT) laser can be described as a transition from laminarto turbulent flows. We demonstrate that in this strongly dissipative system, the transition happenseither via an absolute or a convective instability depending on the laser parameters. In the lattercase, the transition occurs via formation of localised structures in the laminar regime, whichtrigger the formation of growing and drifting puffs of turbulence. Experimentally, we demonstratethat these turbulent bursts are seeded by appearance of Nozaki-Bekki holes, characterised by thezero field amplitude and π phase jumps. Our experimental results are supported with numericalsimulations based on the delay differential equations model.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

OCT is a powerful imaging technique that enables the acquisition of real-time images of scatteringmedia that are semi-transparent in the near-infrared range [1]. While OCT initially relied onwhite light interferometry using low coherence superluminescent diodes, new techniques suchas swept-source OCT (SS-OCT) enable the realisation of 3D-real-time video of biologicaltissues [2]. In SS-OCT, the image quality depends on the property of a swept source, which is alaser whose output wavelength is periodically modulated [3]. For example, the axial resolutionis inversely proportional to the width of the optical spectrum while the image depth is limitedby the coherence length. This coherence length, which can be measured by a conventionalinterferometric technique or by a more complex direct measurement of the electric field [4], islimited by various physical phenomena ranging frommode-hopping [5], parasitic noise source [6],cavity dispersion [7] or, more generally, by the appearance of instabilities. In this paper, we studycoherence properties of a long SS-OCT laser and find out that the mechanism leading to the lossof coherence is similar to the laminar-turbulent transition in hydrodynamic flows.

The laminar-turbulent transition [8,9] is a crucial problem of engineering and science motivatedby many applications such as turbulent transport, atmospheric flows and airplane developmentwhile remaining a partially understood problem of fluid dynamics [10, 11]. The transition fromlaminar to turbulent flow follows usually the same scenario, evolving from a stationary motionto well-defined oscillations eventually ending in a turbulent wake. For example, the Poiseuilleflow inside a pipeline demonstrates an interesting example of the transition to turbulence since

#348383 https://doi.org/10.1364/OE.27.016395 Journal © 2019 Received 25 Oct 2018; revised 26 Mar 2019; accepted 1 Apr 2019; published 24 May 2019

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the laminar flow is theoretically linearly stable for any speed [12] (or Reynolds number) butexperimental observations show the development of turbulence associated with the formationof a turbulent spot downstream indicating that a finite amplitude perturbation destabilises thelaminar regime [9].The important role of nonlinearities was first understood phenomenologically in the 1950s

after the pioneering work of L. D. Landau [13]. The transition from a laminar to a turbulentstate may be characterized by the notion of bifurcation which has been at the centre of importantdevelopments from the 60s up to the present [14]. Many features observed in a turbulentflow are closely connected to the formation of localised structures such as solitary waves orvortices. These structures are naturally observed in nonlinear optics [15, 16], which has becomean excellent test-bed to investigate turbulence and the emergence of complexity in spatiallyextended systems [17]. It was, for example, suggested that the transition to turbulence in a longcavity laser could be explained in the frame of a nearly conservative system where the balancebetween dispersion and Kerr nonlinearities leads to dark and grey soliton clustering [18]. Roguewaves [19] and turbulent puffs in the transition to turbulence [20] have also been observed inpolarisation dynamics of partially mode-locked and quasi-CW lasers.Here, we demonstrate that the transition from laminar to turbulence can occur in a long laser

operating at the zero dispersion point. In addition to dissipative saturable nonlinearities, ourlaser system includes a tunable spectral filter. The spectral filtering is equivalent to diffusionwithin the framework of the complex-Ginzburg Landau equation [21]. If the filter is driven innear resonance with the cavity round trip time, we observed and analysed the transition betweenabsolute and convective instabilities. In the convective instability regime, we show that turbulentpuffs, generated by localised structures, seed the growth of turbulence leading to de-coherence ofthe laser field. Convective instabilities are commonly observed in open flow instabilities wherea disturbance may grow in time as it travels away from the region of its birth so its amplitudegrows at any point along the path before eventually decaying to its original value. This contrastswith the more conventional absolute instabilities where an initial disturbance spreads out alongthe entire unstable region [22–25]. In the convective regime, noise-induced spots, swirls andother structures are commonly observed in hydrodynamic turbulence. Noise may also lead tothe appearance of a laminar-turbulent transition at finite distance in the convective regime asdemonstrated by Deissler [26]. Turbulent puffs in the transition to turbulence have also beenobserved in partially mode-locked laser [20] and associated with the formation of rogue waves. Inour case, turbulent puffs are generated by Nozaki-Bekki holes which we observed experimentallyand analyzed in the frame of delay differential equations.

2. Experimental setup and theoretical model

Our experimental setup consisted of a long cavity swept laser which incorporated a semiconductoramplifier as a gain medium and a narrow bandwidth (50pm) Fabry-Pérot tunable filter. Theoptical output of the laser was following the central frequency of the filter (∆(t)) generatingwavelength sweeps of several nm around the 1300nm central wavelength. The temporal evolutionof the output power was studied by coupling the laser light to a high speed photodetector andfurther analysed on a 12GHz real-time oscilloscope. The laser cavity had a length of about20m. In this case, the laser was operating in a quasi-static regime meaning the frequency of thewavelength sweep was in the order of 1Hz.

Theoretically, we analysed the system using the following delayed differential equations thatdescribe the temporal evolution of the laser gain G(t) and the electric field envelope A(t) asin [27]. These equations read

ÛA(t) + (Γ − i∆ (t)) A(t) = Γ√κe(1−iα)G(t−T )/2 A (t − T) , (1)

ÛG(t) = γ[g0 − G(t) −

(eG(t) − 1

)|A(t)|2

], (2)

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where ÛA(t) and ÛG(t) are the time derivatives of the electric field envelope and the laser gain, Tis the delay time, corresponding to the roundtrip time of the light in the cavity, g0, γ, α, Γ andκ are the unsaturated gain, gain relaxation rate, linewidth enhancement factor, filter width andattenuation factor per roundtrip from cavity losses, respectively.

(a)

(b)

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λfilte

r(a

.u.)

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0

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nsity (

a.u

.)

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Fig. 1. Subcritical and supercritical bifurcations in awavelength swept laser. (a) Experimentalset-up of a long cavity wavelength swept laser incorporating a semiconductor optical amplifier(SOA) and a fast Fabry-Pérot tunable filter (FFP-TF) (see the detailed description of thelaser components in [27]). (b) The filter wavelength variation in time (top). Experimental(middle) and theoretical (bottom) observations of subcritical and supercritical modulationalinstabilities. A sequence of subcritical bifurcations occurs as the filter transmissionwavelength increases while a single supercritical bifurcation leads to the appearance of aturbulent state as the the filter transmission wavelength decreases. The experimental timetrace was recorded for the laser with a 20m cavity length and 1Hz filter modulation frequency.The theoretical time trace was obtained by direct numerical simulation of Eqs. (1) and (2)with a cavity round trip time of 14ns.

When the filter moved in a low sweep rate regime, we observed both experimentally andtheoretically, that the single mode laser operation underwent either a subcritical or a supercriticalHopf bifurcation depending on the direction of the filter tuning as described in [27]. The increaseof the filter transmission wavelength resulted in a subcritical bifurcation that ultimately led to thetransition to another, red-detuned, single mode operation. The decrease of the filter transmissionwavelength resulted in a supercritical bifurcation which led to the emergence of a turbulentregime. Such behaviour, as illustrated in Fig. 1(b), was observed when the filter wavelength wasperiodically modulated at a very low frequency (less than 10Hz).

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Next, the laser was set to operate in a Fourier Domain Mode-Locked (FDML) regime [28].This regime is achieved by adding extra single mode fiber delay into the cavity and increasingthe cavity length up to 20km. In this case it was possible to synchronise the period of the filtermodulation (TF ) with the roundtrip time of the light in the cavity (T). An interpretation of thephase shift introduced by dynamically tuned Fabry-Pérot filter with the help of Doppler effectis given in [29,30]. In this regime, the sweeping rate was in the order of 10kHz. Interestingly,the motivation to investigate the dynamics of FDML lasers is mostly driven by its applicationin SS-OCT [1, 2, 31] and more recently, Raman spectroscopy [32]. This laser, due to itsunprecedented high sweep rates, enables volumetric video-rate real time OCT imaging whosequality relies on ultra-fast broadband swept sources with a long coherence length. However theloss of coherence within the sweep leads to deterioration of the image quality and limits thegeneration of extremely short pulses to 60ps after optical decompression [33].

The bifurcation analysis of the FDML laser reveals similar regimes as described in the case ofa low sweep rate laser [27]. In order to relate the evolution of the FDML output to the turbulentbehaviour commonly observed in spatio-temporal systems, we reconstructed space-time diagramsof the laser intensity in analogy with [34].

Experimentally, this was carried out by recording the laser output during a small time windowof length tW (tW � T), repeated with a period matching the filter sweep period TF . For eachfilter sweep number n, we examined the intensity of the laser in the interval nTF < t < nTF + tW .The electric field was then treated as a function of two variables: time and filter period number n.A 2D picture of the laser dynamics is thus created for a small part of the filter sweep.

In our theoretical simulations, we followed a similar approach, integrating in the limit of avery large delay near the bifurcation point. The experimental realisation of the FDML regimerequired a very long cavity length. The delay differential equation model could be used to runsimulations of this laser but the delay would be very large, requiring lots of computation. Thedynamics of interest are in a small time window where laminar flow drifts into a region ofinstability and becomes turbulent over many sweep periods. The simplest case is close to thefilter turning point, where the filter moves slowly and the laminar flow is a single mode solutionto the delayed differential equations model, instead of a chirped FDML solution. By using theanalytical solution for the laminar part and by assuming the cavity roundtrip time to be arbitrarilylong, we can simulate only the required portion of each filter sweep (Table 1).

Table 1. Parameter values for simulations

Parameter Description Value

γ Carrier relaxation rate 4GHz

κ Linear attenuation factor 0.04

α Linewidth enhancement factor 5.0

g0 Unsaturated gain per roundtrip 5.0

πΓ Filter full width at half maximum 9GHz

The electric field during the nth sweep is An(t) = A(t + nTF ). We consider the case whereT = TF + τ, with τ > 0, so that the electric field drifts in the positive t direction relative to thefilter sweep profile with each subsequent period. In order to simulate An(t) in the time window0 < t < tW , the required delayed term is given by An−1(t), with the time window offset by τ asshown in Fig. 2. For t > 0 this history comes from previous simulation or initial conditions. Weassume that the filter is static at zero frequency for an arbitrarily long time before t = 0 so that a

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single mode solution can be calculated analytically in this region, filling in the history for t < 0.

Δ t)

τ

n 1 TF

tW

n TFt

Fig. 2. Plot of filter sweep profile, ∆(t), used for simulations. For each sweep period n themodel equations are numerically integrated in the green shaded region using the delayedterm from the blue shaded region. The solution in the region where ∆(t) = 0 is calculatedanalytically and the dashed region where the filter returns to zero is ignored.

The full filter sweep used for simulation is as follows. The filter starts at frequency f0 at t = 0and decreases linearly during the simulated time interval. After the simulated interval the filterreturns to f0 and remains there for the remainder of the arbitrarily long filter sweep, allowing thesystem to return to single mode operation before the beginning of the next simulated interval.The filter sweep profile of the real laser follows a sine wave. Near the turning point the filter isalmost static for a time which is long compared with the time scale of the laser dynamics.There is some equivalence between the single mode operation and chirped FDML operation.

The following transformation puts the system in a frame where the filter is always at zerofrequency, A(t) = a(t) exp

(i∫ t

t−T ∆(t′)dt ′

). If the filter period is perfectly tuned to match the

cavity round trip time then this system is identical to a system with a static filter, but the singlemode solutions in this “filter frame” system correspond to chirped FDML solutions. In the caseof small detuning these solutions drift away from the filter over many round trips. LaminarFDML thus has similar instabilities to single mode operation, so a study of the single mode caseprovides insight into the transition to turbulence in FDML.

3. Nozaki-Bekki holes

When the filter was driven in a perfect resonance with the cavity roundtrip time, the laser emitteda set of frequency modulated outputs corresponding to single modes in the filter frame [27].These modes have the same stability criteria as the modes observed when the central frequencyof the filter is fixed. When the filter was tuned out of resonance from the cavity round trip, thestability of these frequency modulated outputs depended on detuning δ = 2π(1/TF −1/T). Whilethe behaviour of the supercritical bifurcation remained qualitatively the same for any values of δ,the dynamics around the subcritical point strongly depended on its value and, to investigate thispoint further, we considered the dynamics for the increasing values of δ > 0. As δ increased,we identified four different regimes. For δ < δc1 , the laminar regime remained stable for theentire sweep. For δc1 < δ < δc2 , the laminar regime became turbulent as soon as it passed thebifurcation point; the transition between laminar and turbulent regimes remained at a fixed valueof the filter frequency. This regime is consistent with an absolute instability as illustrated by 2Ddiagrams in Figs. 3(a) and 4(a) (experimental) and Figs. 3(b) and 4(b) (theoretical).For the larger detuning values, i.e. δc2 < δ < δc3 , the laminar regime invaded the turbulent

regime at a rate much faster than the growth rate of the turbulent regime (Figs. 3(c) and 4(c)(experimental) and Figs. 3(d) and 4(d) (theoretical)). In this case, the front between the laminarand turbulent regimes was no longer synchronised with the filter position but returned at everyroundtrip time. In the filter frame this appeared as a front drifting with a “speed” proportional to

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Fig. 3. Evolution of the laser intensity near the transition between the laminar and turbulentregimes for subsequent filter periods (n). Experimental (a) and theoretical (b) 2D diagramsof the laser intensity evolution for 250 subsequent roundtrips in the absolute instabilityregime corresponding to δc1 < δ < δc2 . Experimental (c) and theoretical (d) 2D diagramsof the laser intensity evolution for the subsequent 500 roundtrips in the convective instabilityregime corresponding to δc2 < δ < δc3 . The triangular features in (c) and (d) represent theemergence of localised structures from the laminar regime. These structures drift towardthe turbulent region with a speed v ∼ 1/pup and the size of the turbulent spot grows ata rate r ∼ 1/plow − 1/pup , where pup and plow are the slopes of the upper and lowerboundaries between the laminar state and the localised structures. The colorbars representthe normalised laser intensity. The parameters for simulations are given in Table 1.

the detuning. As the laminar regime invaded the turbulent region, we observed spontaneous noiseinduced creation of turbulent puffs which were mediated by the formation of spatio-temporaltopological defects. For δc3 < δ, the laser displayed a turbulent regime throughout the entiresweep. Experimentally, the values for δc1 , δc2 were found to be on the order of a few mHz whileδc3 was on the order of a few hundreds of mHz. The values of detuning below δc1 are the mostpromising for the imaging applications, yet difficult to achieve experimentally [29].

The temporal evolution of the laser intensity near the subcritical bifurcation point as a functionof the filter period number n is also shown in Figs. 4(a)–4(d). Figures 4(a) and 4(b) demonstratethe sharp transition and Figs. 4(c) and 4(d) depict the regime where convective instability occursdisplaying the drift of the laminar regime in the turbulent regime and the creation of localisedstructures. For example, Fig. 4(c) shows the experimentally observed formation of a localisedstructure near t = 10ns for n = 220.

To further characterise the formation of the turbulent puffs, we measured both the intensity and

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(a) (c)

n=0

n=50

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n=200

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n=10

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nsity (

a.u

.) n=40

n=70

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n=110

Fig. 4. The detailed analysis of the laser intensity evolution near the subcritical bifurcationpoint. The laser intensity was recorded at t = nTF , where n is the number of the filtercycles of modulation. Experimental (a) and theoretical (b) observations of the laser intensitydemonstrating a sharp transition from laminar to turbulent regimes at a small value ofdetuning δc1 < δ < δc2 . This corresponds to the absolute instability regime. The animationsof the corresponding regime are shown in Visualisation 1 (experiment) and Visualisation2 (simulations). Experimental (c) and theoretical (d) observations of creation and drift ofturbulent fronts in the convective regime (δc2 < δ < δc3 ). The nucleation of a hole structurethat appears suddenly in the cavity is shown in (c) at round trip 220. The animations ofthe corresponding regime are shown in Visualisation 3 (experiment) and Visualisation 4(simulations). The parameters for simulations are given in Table 1.

phase of the laser in the initial stage of the creation of these localised structures. To measure thephase, we used the experimental set-up described in [4, 35]. In short, the set-up included a 3x3self-delayed heterodyne that enabled us to measure the phase difference φ(t) − φ(t − Td) = η(t),where Td is the time difference between the two inputs of the 3x3 coupler. The measurement ofthe temporal evolution of the laser intensity and phase with a 25ps temporal resolution showsthat the laser intensity reaches zero with an associated π-phase jump (Fig. 5(a)). Similar results,obtained numerically, are shown in Fig. 5(b). The 2D diagrams of the laser intensity and theassociated real part of the electric field are also shown on Fig. 6(b). It is also worthwhile tonote that such interferometric technique enabled us to measure the laser linewidth in the variousregimes as described in [6]. Using this technique, we observed a coherence collapse at thelaminar-turbulent transition where the linewidth on the order of 500MHz in the laminar regimeincreased to the linewidth of a few GHz in the turbulent regime. This coherence collapse is

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another indicator of the growth of turbulence.

(a) (b)

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nsity(a

.u.)

0 1 2 3

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Rad)

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nsity(a

.u.)

0 1 2 3

Time (ns)

-1

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Ph

ase

(π

Ra

d)

Fig. 5. Experimentally observed (a) and, theoretically modelled (b), temporal evolutionof the laser intensity (top) and phase (bottom) during the creation of a localised structure.Note that the laser intensity vanishes while the phase exhibits a π-jump as observed withNozaki-Bekki holes.

(a) (b)

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Fig. 6. Creation of three turbulent puffs. Experimentally measured temporal evolution of thelaser intensity (a), and real part of the laser field (b) for 250 subsequent filter periods showingthe emerging of three Nozaki-Bekki holes that initiate three turbulent regions within thelaminar regime.

The observed localised structures (Fig. 5) are similar to the Nozaki-Bekki holes [36, 37]observed in complex Ginzburg-Landau equation. These holes are asymmetric wave sources,emitting different wave numbers up and down streams [38], and it has been suggested that theyare the building-blocks of spatio-temporal chaos in convection experiments [39]. In our case,the holes move “downstream” and nucleate the meta-stable turbulent phase that propagates ata larger speed, as demonstrated experimentally and numerically in Figs. 4(c) and 4(d). Theseturbulent regions grow while drifting and merging with other turbulent regions. As a result, thelaminar region continues to invade the turbulent region and other turbulent domains are created.

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4. Conclusion

In summary, we have provided, both experimentally and numerically, a scenario for the lossof coherence in a long cavity swept laser. By precisely controlling the detuning between thecavity roundtrip time and the sweep period we were able to demonstrate the transition betweenan absolute and convective instabilities. In the convective instability regime, we observe atransition to turbulence via creation of convective Nozaki-Bekki holes. This optical analogue ofturbulent puffs in hydrodynamics, well-known as Poiseuille flow, reveals the mechanism thatleads to deterioration of the coherence of the laser. Delayed differential equations modeling is inexcellent qualitative agreement with the experimental observations. It is, to our knowledge, thefirst experimental observation of Nozaki-Bekki holes inducing the turbulent transition in opticalsystems.

Funding

Marie Sklodowska-Curie Individual Fellowships European H2020-MSCA-IF-2017, ICOFASproject (800290); Science Foundation Ireland (11/PI/1152); OPTIMAL project granted bythe European Union by means of the Fond Européen de développement regional, FEDER;Government of Russian Federation (08-08); Fédération Doeblin CNRS, SFB 787 of the DFG(project B5).

Acknowledgments

The authors thank Stephen Hegarty for help with the experiment and Natalia Rebrova for usefuladvice and encouraging discussions.

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